# Properly interpreting propositions 2.5 and 2.6 in Lee's Introduction to Smooth Manifolds

I want to make sure that I am correctly interpreting propositions $$2.5$$ and $$2.6$$ (picture below) of Lee's Introduction to Smooth Manifolds. Are the two following interpretations correct?

• In $$2.5$$ (b) the smooth atlases $$\{(U_\alpha,\phi_\alpha)\}$$ and $$\{(V_\beta,\psi_\beta)\}$$ are taken to be subsets of the smooth (maximal) atlas $$\mathcal{A}$$ that gives $$M$$ its smooth manifold structure.

• In $$2.6$$ the subset $$U$$ is interpreted as a smooth submanifold of $$M$$, so as to be capable of speaking of $$F|_U$$ being smooth. • first one says that you dont have to check every single atlas. The second basically says that if you are locally smooth, then you are smooth (this is not always true such as the property of being an embedding) Aug 30 at 0:26

Good try! Your interpretation isn’t quite right. Given a fixed atlas, you can construct a maximal atlas “by adding to it every compatible chart”. In this case, $$\{U_\alpha, \phi_\alpha\}$$ and $$\{V_\beta, \psi_\beta\}$$ are atlases of two different manifolds, $$M$$ and $$N$$, respectively.
This is a very simple “definition chase”. Let’s prove (a) implies (b). Since $$M$$ and $$N$$ are manifolds, they have fixed atlases $$\{U, \phi\}$$ and $$\{V, \psi\}$$. If (a) holds, then for every point $$p$$, there exist charts (of the fixed atlases) containing $$p$$ and $$F(p)$$ such that the stated triple composition is smooth. Trivially, setting $$\{U, \phi\}$$ = $$\{U_\alpha, \phi_\alpha\}$$, and $$\{V, \psi\}$$ = $$\{V_\beta, \psi_\beta\}$$, we get that (b) holds. Conversely, to show (b) implies (a), you do the definition chase the other way around, it should be easy.
The requirement that $$F$$ is continuous in (b) is equivalent to saying that the pre-image of every open set is itself open. So trivially, the requirement that $$F$$ is continuous in (b) implies that $$F^{-1}(V)$$ is open in $$M$$ and hence $$F^{-1}(V)\cap U$$ is open in $$M$$. Conversely, we need to check $$F^{-1}(V)$$ is open in $$M$$ for each $$V$$ in the atlas of $$N$$, using the condition in (a). Starting from all the conditions in (a), instead of setting $$\{V_\beta, \psi_\beta\}\equiv\{V, \psi\}$$ , we set $$\{V_\beta, \psi_\beta\}$$ to be the maximal atlas containing $$\{V, \psi\}$$. Then, it is a lemma that the maximal atlas contains a basis for the topology of $$N$$ (which, in broad strokes, follows from the fact that the maximal atlas contains the so-called “coordinate balls” in Lee’s terminology). It is a fact that it suffices to check that the pre-image of every open set of a basis for the topology of $$N$$ is open, in order to establish that $$F$$ is a continuous function. So we can check $$F^{-1}(V)$$ is open in $$M$$ for every $$V$$ in the maximal atlas. But we can see that $$F^{-1}(V)$$=$$F^{-1}(V)\cap M$$= $$F^{-1}(V)\cap (\cup_i U_i)$$ = $$\cup_i (F^{-1}(V)\cap U_i)$$ which is a union of open sets by the condition in (a) that we are assuming. Hence, $$F$$ is continuous as (b) requires.
In 2.6, you don’t need to endow each $$U$$ with a submanifold structure. It can be a single-chart manifold and you can check smoothness directly. In terms of the interpretation, it is telling you that smoothness is a local condition. An easier example to think about is a function from the reals to the reals which is globally $$C^1$$, by definition this means that it the derivative exists and is continuous at every point in the reals. It is trivial to see that globally $$C^1$$ implies locally $$C^1$$, and vice versa.